A monopolist has inverse demand function, $P(Q)$ where $P$ is price and $Q$ is output. The cost per unit of output is $z.$

$($a$)$ Write down the profit function and use it to find the profit maximizing first$-$order condition.

$($b$)$ Assume the first$-$order condition is $P(Q^{**})+Q^{**}{P}^{\text{'}}(Q^{**})=z$ and recognize that the profit maximizing quantity, $Q^{**},$ is an implicit function of $z.$ Use implicit differentiation to show how $Q^{**}$ is affected by changes in cost per unit, $z.$